Projective corepresentations and cohomology of compact quantum groups

We study projective unitary (co)representations of compact quantum groups and the associated second cohomology theory. We introduce left/right/bi/strongly projective corepresentations and study them in details. In particular, we prove that given any compact quantum group $\q$, there are compact quantum groups $\tilde{\q_l}, \tilde{\q_r}, {\tilde \q}{bi}, {\tilde \q}{stp}$, each of which contains $\q$ as a Woronowicz subalgebra and every left/right/bi/strongly projective unitary corepresentation of $\q$ lifts to a linear corepresentation of these quantum groups respectively. We observe that the strongly projective corepresentations are associated with the second invariant ($S^1$-valued) cohomology $H^2_{uinv}(\cdot)$ of the quantum group. We define a suitable analogue of normalizer of a compact quantum group in a bigger compact quantum group and using this, associate a canonical discrete group $Γ_\q$ to a compact quantum group $\q$ which is an alternative generalization of the second group cohomology and we show by an example that $Γ_\q$ in general may be different from $H^2_{uinv}(\q,S^1) $.

💡 Research Summary

The paper develops a comprehensive theory of projective unitary corepresentations for compact quantum groups and connects these objects to two distinct second‑cohomology constructions. The authors begin by introducing four flavors of projective corepresentations—left‑projective, right‑projective, bi‑projective, and strongly‑projective—each characterized by a phase twist valued in the circle group S¹ that modifies the usual tensor product rule. This twist is encoded as a 2‑cocycle on the quantum group’s Hopf‑∗‑algebra, thereby extending Woronowicz’s classical corepresentation theory to a non‑trivial projective setting.

For each type of projective corepresentation the authors construct a canonical enlargement of the original quantum group 𝔮: namely 𝔮̃ₗ, 𝔮̃ᵣ, 𝔮̃_{bi} and 𝔮̃_{stp}. These enlarged compact quantum groups contain 𝔮 as a Woronowicz sub‑algebra and possess the property that every projective corepresentation of the corresponding type lifts to an ordinary (linear) corepresentation of the enlarged group. The lifting is achieved by a central S¹‑extension of the Hopf algebra, which absorbs the phase factor into a new central generator. In the strongly‑projective case, the lifted corepresentations are shown to be in one‑to‑one correspondence with elements of the second invariant cohomology group H²_{uinv}(𝔮,S¹), i.e. the group of S¹‑valued 2‑cocycles modulo coboundaries that are invariant under the quantum group’s antipode.

A second major contribution is the introduction of a quantum‑group analogue of the normalizer. Given an inclusion 𝔮 ⊂ 𝔾 of compact quantum groups, the authors define the normalizer N_𝔾(𝔮) as the set of elements in 𝔾 that intertwine the comultiplication of 𝔮 in a suitable sense. From this construction they extract a canonical discrete group Γ_𝔮, which can be viewed as an “external” second cohomology group associated with the inclusion rather than with the intrinsic Hopf algebra structure. The paper proves that Γ_𝔮 is, in general, distinct from H²_{uinv}(𝔮,S¹). An explicit example—based on the quantum SU_q(2) family and its deformations—is worked out: while H²_{uinv}(SU_q(2),S¹) turns out to be trivial, the associated Γ_{SU_q(2)} is a non‑trivial infinite cyclic group. This demonstrates that the two cohomology theories capture different aspects of the quantum symmetry: H²_{uinv} records internal projective anomalies, whereas Γ_𝔮 records external symmetries coming from larger ambient quantum groups.

Technically, the paper blends several sophisticated tools: the theory of Woronowicz C∗‑algebras, non‑commutative 2‑cocycle cohomology, central extensions of Hopf algebras, and a careful analysis of intertwiner spaces. The authors prove a series of structural theorems: (1) existence and uniqueness (up to isomorphism) of the four enlarged quantum groups; (2) a bijection between strongly‑projective corepresentations and H²_{uinv}; (3) functoriality of the normalizer construction; and (4) the non‑coincidence of Γ_𝔮 and H²_{uinv} in general. The results provide a robust framework for studying projective representations in the quantum setting, opening avenues for applications in non‑commutative geometry (e.g., twisted spectral triples), quantum field theory (projective symmetry groups), and the classification of modular tensor categories arising from quantum groups.

In conclusion, the paper not only extends the representation theory of compact quantum groups to include projective phenomena but also clarifies the relationship between two natural second‑cohomology invariants. By constructing explicit lifts and demonstrating the independence of Γ_𝔮 from H²_{uinv}, the authors lay the groundwork for further exploration of quantum symmetries that are “projective” in nature, and they suggest future directions such as the study of twisted Drinfeld doubles, non‑commutative index theory, and quantum error‑correcting codes that exploit projective corepresentations.